
But the quantities are not conserved for differences of solutions or \(v\) . Still we can control \(M(v)+E(v)\)  by Grønwall's argument for sufficiently regular initial data like in [1]}, [2]} in the context of the defocusing nonlinear Schrödinger equation.
In the proof we have to control \(\Vert  w(t) \Vert _{L^6(\mathbb {R}^2)}\)  and \(\Vert  w(t) \Vert _{L^\infty (\mathbb {R}^{d})}\) , for which we use embeddings, namely (REF ), Propositions REF  and REF , and a standard Sobolev embedding:
\(\Vert  w(t) \Vert _{L^6(\mathbb {R}^2)} &\lesssim \Vert  w(t) \Vert _{\mathcal {B}^{1/6}_{6,6,6}(\mathbb {R}^{2})} \lesssim \Vert  w(t) \Vert _{\mathcal {B}^{1/6+\varepsilon }_{6,2,2}(\mathbb {R}^2)}, \\\Vert  w(t) \Vert _{L^\infty (\mathbb {R}^2)} &\lesssim \Vert w(t) \Vert _{W^{1/3+\varepsilon ,6}(\mathbb {R}^2)} \lesssim \Vert  w(t) \Vert _{\mathcal {B}^{1/2+2\varepsilon }_{6,2,2}(\mathbb {R}^2)}.\) 
